CM3. Generation of unidirectional composite SVEs from micro-structural statistical information. Computational & Multiscale Mechanics of Materials

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1 Computational & Multiscale Mechanics of Materials CM3 Generation of unidirectional composite SVEs from micro-structural statistical information Wu Ling, Bidaine Benoit, Major Zoltan, Nghia Chnug Chi, Noels Ludovic The research has been funded by the Walloon Region under the agreement no STOMMMAC (CT-INT ) in the context of the M-ERA.NET Joint Call CM3 CMCS November 2017, Paris, France

2 The problem Material uncertainties affect structural behaviors Probability Probabilistic homogenization Probability E Fibers Probability UQ w I w 0 a Composite stiffness Homogenized material properties distribution E matrix Loading Probability Fiber orientation (a) Probability Stochastic structural analysis Failure load CM3 CMCS November 2017, Paris, France - 2

3 s [Mpa] The problem Illustration assuming a regular stacking 60%-UD fibers Damage-enhanced matrix behavior d min ϴ Effect of the loading direction q for d min = 0.5 µm Effect of the distance d min for q = 30 o 80 0 degree 80 s [Mpa] degree 30 degree 45 degree 60 degree 75 degree dmin=0.2 dmin=0.35 dmin=0.5 dmin= e 90 degree e dmin=0.8 Question: what does happen for a realistic fibre stacking? CM3 CMCS November 2017, Paris, France - 3

4 Experimental measurements 2000x and 3000x SEM images Fibers detection CM3 CMCS November 2017, Paris, France - 5

5 Micro-structure stochastic model Basic geometric information of fibers' cross sections Fiber radius distribution p R r Basic spatial information of fibers The distribution of the nearest-neighbor net distance function p d1st d The distribution of the orientation of the undirected line connecting the center points of a fiber to its nearest R 2 neighbor p θ1st θ θ 2nd The distribution of the difference between the net distance to the second and the first nearest-neighbor p Δd d with Δd = d 2nd d 1st d 2nd θ R 0 d 1st θ 1st R 1 The distribution of the second nearest-neighbor s location referring to the first nearest-neighbor p Δθ θ with Δθ = θ 2nd θ 1st CM3 CMCS November 2017, Paris, France - 6

6 Micro-structure stochastic model Histograms of random micro-structures descriptors R 2 θ 2nd d 2nd θ d 1st R 1 θ 1st R 0 CM3 CMCS November 2017, Paris, France - 7

7 Micro-structure stochastic model Dependency of the four random variables d 1st, d, θ 1st, θ Correlation matrix d 1st d θ 1st θ R 2 d 1st θ 2nd d d 2nd θ d 1st R 1 θ 1st θ 1st θ 1.0 R 0 Distances correlation matrix d 1st and d are dependent they will be generated from their empirical copula d 1st d θ 1st θ d 1st d θ 1st θ 1.0 CM3 CMCS November 2017, Paris, France - 8

8 Micro-structure stochastic model d 1st and d should be generated using their empirical copula SEM sample Generated sample d d Directly from copula generator R 2 d 1st d 2nd θ θ 2nd d 1st R 1 d Statistic result from generated SVE θ 1st R 0 d 1st CM3 CMCS November 2017, Paris, France - 9

9 Micro-structure stochastic model The numerical microstructure is generated by a fiber additive process 1) Define N seeds with first and second neighbors distances d 2ndk d 1stk Seed k R k d 2ndk+1 d 1stk+1 Seed k + 1 R k+1 CM3 CMCS November 2017, Paris, France - 10

10 Micro-structure stochastic model The numerical microstructure is generated by a fiber additive process 1) Define N seeds with first and second neighbors distances 2) Generate first neighbor with its own first and second neighbors distances Seed k d 2nd0 R 0 d 1st0 θ 1st d 2nd1 R 1 d 1st1 d 2nd0 d 1st0 Seed k + 1 R 0 CM3 CMCS November 2017, Paris, France - 11

11 Micro-structure stochastic model The numerical microstructure is generated by a fiber additive process 1) Define N seeds with first and second neighbors distances 2) Generate first neighbor with its own first and second neighbors distances 3) Generate second neighbor with its own first and second neighbors distances d 2nd2 d 1st2 R 2 Seed k d 2nd0 Δθ R 0 d 1st0 θ 1st d 2nd1 R 1 d 1st1 d 2nd0 d 1st0 Seed k + 1 R 0 CM3 CMCS November 2017, Paris, France - 12

12 Micro-structure stochastic model The numerical microstructure is generated by a fiber additive process d 2ndik d 1stik d 2nd1 1) Define N seeds with first and second neighbors distances R ik d 2nd0 d 1st0 R 1 d 1st1 2) Generate first neighbor with its own first and second neighbors distances 3) Generate second neighbor with its own first and second neighbors distances Seed k R 0 d 2ndik+1 R ik+1 θ 1st d 1stik+1 d 2nd0 R 0 d 1st0 4) Change seeds & then change central fiber of the seeds Seed k + 1 CM3 CMCS November 2017, Paris, France - 13

13 Micro-structure stochastic model The numerical micro-structure is generated by a fiber additive process The effect of the initial number of seeds N and The effect of the maximum regenerating times n max after rejecting a fiber due to overlap SEM: Average V f of 103 windows; Numerical micro-structures: Average V f of 104 windows. CM3 CMCS November 2017, Paris, France - 14

14 Micro-structure stochastic model Comparisons of fibers spatial information R 2 θ 2nd d 2nd θ d 1st R 1 θ 1st R 0 CM3 CMCS November 2017, Paris, France - 15

15 Micro-structure stochastic model Numerical micro-structures are generated by a fiber additive process Arbitrary size Arbitrary number Possibility to generate non-homogenous distributions CM3 CMCS November 2017, Paris, France - 16

16 Stochastic homogenization on the SVEs Stochastic homogenization Extraction of Stochastic Volume Elements 2 sizes considered: l SVE = 10 μm & l SVE = 25 μm Window technique to capture correlation R rs τ = For each SVE Extract apparent homogenized material tensor C M ε M = 1 V ω σ M = 1 V ω C M = E r x E r s x + τ E s E r E r 2 E s E s 2 σ M ω ε m dω ω σ m dω u M M y SVE x, y x SVE x, y t SVE x, y l SVE Consistent boundary conditions: Periodic (PBC) Minimum kinematics (SUBC) Kinematic (KUBC) CM3 CMCS November 2017, Paris, France - 17

17 Stochastic homogenization on the SVEs Apparent properties l SVE = 10 μm l SVE = 25 μm Increasing l SVE When l SVE increases Average values for different BCs get closer (to PBC one) Distributions narrow Distributions get closer to normal CM3 CMCS November 2017, Paris, France - 18

18 Stochastic homogenization on the SVEs When l SVE increases: marginal distributions of random properties closer to normal l SVE = 10 µm l SVE = 25 µm CM3 CMCS November 2017, Paris, France - 19

19 Stochastic homogenization on the SVEs Correlation l SVE = 10 μm l SVE = 25 μm Increasing l SVE (1) Auto/cross correlation vanishes at τ = l SVE (2) When l SVE increases, distributions get closer to normal (1)+(2) Apparent properties are independent random variables However the distribution depend on l SVE The boundary conditions CM3 CMCS November 2017, Paris, France - 20

20 Stochastic homogenization on the SVEs Quid larger SVEs? Computational cost affordable in linear elasticity Computational cost non affordable in failure analyzes How to deduce the stochastic content of larger SVEs? Take advantages of the fact that the apparent tensors can be considered as random variables L BSVE l SSVE CM3 CMCS November 2017, Paris, France - 21

21 Stochastic homogenization on the SVEs Quid larger SVEs? Computational cost affordable in linear elasticity Computational cost non affordable in failure analyzes How to deduce the stochastic content of larger SVEs? Take advantages of the fact that the apparent tensors can be considered as random variables L BSVE l SSVE Level I Level II Computational homogenization CM3 CMCS November 2017, Paris, France - 22

22 Stochastic homogenization on the SVEs Quid larger SVEs? Computational cost affordable in linear elasticity Computational cost non affordable in failure analyzes How to deduce the stochastic content of larger SVEs? Take advantages of the fact that the apparent tensors can be considered as random variables Accuracy depends on Small/Large SVE sizes l SSVE L BSVE l SSVE = 10 μm l SSVE = 25 μm CM3 CMCS November 2017, Paris, France - 23

23 Stochastic homogenization on the SVEs Numerical verification of 2-step homogenization Direct homogenization of larger SVE (BSVE) realizations 2-step homogenization using BSVE subdivisions Level I Level II Computational homogenization CM3 CMCS November 2017, Paris, France - 24

24 Stochastic reduced order model Stochastic model of the anisotropic elasticity tensor Extract (uncorrelated) tensor realizations C M i C M 1 C M 2 C M 3 i Represent each realization C M by a vector V of 9 (dependant) V (r) variables Generate random vectors V using the Copula method Simulations require two discretizations Random vector discretization Finite element discretization CM3 CMCS November 2017, Paris, France - 25

25 Ply loading realizations Non-uniform homogenized stress distributions Different realizations yield different solutions Stochastic reduced order model σ M xx [Mpa] σ M xx [Mpa] σ M xx [Mpa] σ M xx [Mpa] CM3 CMCS November 2017, Paris, France - 26

26 Conclusions Stochastic generator based on SEM measurements of unidirectional fibre reinforced composites Computational homogenization on SVEs Two-step computational homogenization for Big SVEs Future work: nonlinear and failure analyzes CM3 CMCS November 2017, Paris, France - 31

27 Thank you for your attention! CM3 CMCS November 2017, Paris, France - 32